What really is the Maxwell Stress Tensor? I understand that it's derived from $$\mathbf {F} = \int _V ( \mathbf E + \mathbf v \times \mathbf B )\rho \ d \tau$$
Griffiths describes this as "total EM force on the charges in the volume $\mathcal V$".
$$T_{i j} = \epsilon_0 \left(E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + \frac{1}{\mu_0} \left(B_i B_j - \frac{1}{2} \delta_{ij} B^2\right)$$
This leads us the the stress tensor, but there's something that I don't understand. The description given is
Physically, $T$ is the force per unit area acting on the surface.
What surface are we speaking about here? An arbitrary surface? In the case of example 8.2 (net force on a uniformly charge sphere's upper hemisphere), the surface in question is clearly the boundary of the upper hemisphere and it's "disk" separating the two hemispheres. In other cases, such as problem 8.4, where we have two point charges separated by a distance, we must integrate over a particular surface. For such a problem, we must "integrate the stress tensor over the plane equidistant from the two charges", but why? How would summing up the force on the plane separating the two point charges equal the force on each charge?
How could there be "force per unit area" across an empty plane?
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